I would cut so that the blue borders were entirely removed & then make the orange part as close to squared up as possible (meaning characters may not be exactly straight, but I think that's part of the charm of cartoons).
If you aren't working on straight of grain, make sure to use LOTS of starch/sizing -- or as luvspaper suggested, add a stabilizer. Trying to add piecing onto a bias cut can bring even more problems. Tear away stabilizer would probably be best, but I rarely use it. Typically if I HAD to work on a bias edge, I would just completely saturate the fabric before cutting to ensure it's stiff as paper. You might also consider using a pinking blade on your rotary cutter to help reduce fraying on the bias.